%I A081621
%S A081621 0,0,0,0,0,0,0,0,1,0,1,1,3,4,12,23,73,192,651,2070,7290,25381,91441,
%T A081621 329824,1204737,4412031,16248772,59995535,222231424,825028656,
%U A081621 3069993552,11446245342,42758608761,160012226334,599822851579,2252137171764,
               8469193859271
%N A081621 Number of n-node triangulations of the sphere with minimal degree 5.
%C A081621 Other face sizes bigger than 5 and 6 are allowed and there can be more 
               than 12 vertices with degree 5.
%C A081621 Convex polytopes with minimum degree at least 5. The sequence is extracted 
               from the file more-counts.txt that comes with the plantri distribution.
%C A081621 Grace conjectured that all polyhedra inscribed in the unit sphere with 
               maximal volume are "medial" (all faces triangular and vertex degree 
               either m or m+1 where m<6-12/n<m+1). For n=12 and n>13 the medial 
               polyhedra have 12 vertices of degree 5 and n-12 vertices of degree 
               6. All known numerical solutions of the maximal volume problem (A081314) 
               have this property.
%C A081621 The triangulated arrangements of points on a sphere with icosahedral 
               symmetry given by Hardin, Sloane and Smith are examples for large 
               n.
%D A081621 G. Brinkmann and B. D. McKay, Construction of planar triangulations with 
               minimum degree 5, Discr. Math. 301 (2005), 147-163.
%D A081621 D. W. Grace, Search for largest polyhedra. Math. Comp. 17, 197-199 (1963)
%H A081621 Gunnar Brinkmann and Brendan McKay, <a href="http://cs.anu.edu.au/people/
               bdm/plantri/">Plantri and fullgen</a> programs for generation of 
               certain types of planar graph.
%H A081621 G. Brinkmann and B. D. McKay, <a href="http://cs.anu.edu.au/~bdm/plantri/
               min5paper_sc.pdf">Construction of planar triangulations with minimum 
               degree 5</a>, Discr. Math. 301 (2005), 147-163.
%H A081621 R. H. Hardin, N. J. A. Sloane and W. D. Smith, <a href="http://www.research.att.com/
               ~njas/icosahedral.codes/">Spherical Codes with Icosahedral Symmetry.</
               a>
%H A081621 Hugo Pfoertner, <a href="http://www.enginemonitoring.org/sphere/icoscov.pdf">
               Icosahedral best coverings.</a>
%H A081621 Thom Sulanke, <a href="http://hep.physics.indiana.edu/~tsulanke/graphs/
               surftri/">Generating triangulations of surfaces (surftri)</a>, (also 
               subpages).
%e A081621 With vertices denoted by letters a, b, ... the neighbor lists are for 
               a(14)=1: (bcdef, afghc, abhid, acije, adjkf, aeklgb, bflmh, bgmic, 
               chmnjd, dinke, ejnlf, fknmg, glnih, imlkj)
%e A081621 a(15)=1: (bcdefg, aghic, abijd, acjke, adklf, aelmg, afmhb, bgmni, bhnjc, 
               cinokd, djole, ekomf, flonhg, hmoji, jnmlk); a(16)=3: (bcdef, afghc, 
               abhijd, acjke, adklf, aelmgb, bfmnh, bgnic, chnoj, ciokd, djople, 
               ekpmf, flpng, gmpoih, inpkj, konml), (bcdef, afghc, abhijd, acjke, 
               adklf, aelmgb, bfmnh, bgnic, chnoj, ciopkd, djple, ekpmf, flpong, 
               gmoih, inmpj, jomlk), (bcdef, afghijc, abjkd, ackle, adlmf, aemgb, 
               bfmnh, bgnoi, bhopj, bipkc, cjpld, dkponme, elngf, gmloh, hnlpi, 
               iolkj);
%Y A081621 Cf. A000109, A000103, A081314.
%Y A081621 Sequence in context: A006791 A111358 A111357 this_sequence A073713 A084921 
               A070765
%Y A081621 Adjacent sequences: A081618 A081619 A081620 this_sequence A081622 A081623 
               A081624
%K A081621 nonn
%O A081621 4,13
%A A081621 Hugo Pfoertner (hugo(AT)pfoertner.org), Mar 24 2003
%E A081621 More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007